Asymptotic conjecture for reversible binary prime pairs

Let ${\mathcal B}_n$ be the set of binary integers with $n$ binary digits, let $\overleftarrow{a}$ denote the integer obtained by reversing the binary digits of $a$, and let $\varTheta(n)$ be the number of $a\in{\mathcal B}_n$ such that both $a$ and $\overleftarrow{a}$ are prime. Asymptotic conjecture.

$$\varTheta(n)=(3+o(1))\frac{2^{n-1}}{\left(\log 2^n\right)^2}\qquad(n\to\infty).$$

This heuristic prediction comes from treating primality of a binary integer and of its reversal as conditionally independent after accounting for the shared congruence obstruction modulo $3$. The stated calculations agree with the prediction for all $n\leqslant 50$; no proof of the asymptotic is supplied here.